Mathematical Theory

Español Tuning theory can become heavily mathematical, and here are links to some of the pages which wade into deeper waters.

Scale Basics

• Periodic scale
• MOS scales
• Mathematics of MOS
• MODMOS scales
• Graph-theoretic properties of scales

Regular Temperaments

Basics:

• Monzos and Interval Space
• Vals and Tuning Space
• Regular Temperaments
• Saturation
• Dual list
• Meet and Join

Linear algebra:

• Temperament Mapping Matrices (M-maps)
• Tmonzos and Tvals
• Subgroup Mapping Matrices (V-maps)
• Smonzos and Svals
• Generalized Tenney Norms and Tp Interval Space
• Generalized Tenney Dual Norms and Tp Tuning Space

Exterior algebra:

• Wedgies and Multivals
• The wedgie
• Recoverability
• The dual
• Interior product

Abstract algebra:

• Groups, Lattices, Z-modules and Vector spaces

Temperament representations:

• Abstract regular temperament
• Normal lists
• Gencom

Other:

• Planar Temperament
• Transversal generators

Temperament Measures and Optimization

• Target tunings
• TOP tuning
• Tp tuning
• Tenney-Euclidean Tuning
• Tenney-Euclidean temperament measures
• Tenney-Euclidean metrics
• Cangwu badness
• Fractional monzos
• Generator complexity

Lattices and Scales

• Transversal
• Fokker blocks
• Weak Fokker blocks
• Minkowski reduced bases for Fokker groups of certain vals
• Minkowski blocks
• The Seven Limit Symmetrical Lattices
• Tablets
• Hahn distance
• Comma-based lattices

Other

• The Riemann Zeta Function and Tuning
• Modulatory space
• Chordal space
• k*N subgroups
• Logarithmic approximants
• Don Page comma
• Sane and Insane Temperaments
• ABC, High Quality Commas, and Epimericity